Firms with excess cash face a strategic choice: return it to shareholders as dividends or reinvest into capital projects.

This decision is guided by the cost of capital, the minimum expected rate of return necessary to justify not paying out dividends.

The cost of capital is a benchmark【基准】 representing the returns shareholders expect from investments of similar risk.

Investments should offer a safety margin above the cost of capital to provide a buffer for risk and enhance shareholder value creation.

Understanding the Cost of Equity Capital

The Cost of Equity Capital is what a firm expects to pay to compensate equity investors for the risk they take on:

  • $R_S = R_F + β × (R_M − R_F)$

Estimating this cost requires:

  • Risk-free rate ($R_F$): The return on risk-free securities.
  • Market risk premium ($R_M − R_F$): The additional return expected from the market over the risk-free rate.
  • Stock beta ($β$): The measure of how much the stock’s return moves with the market.

Calculating the Cost of Equity: An Example

Determining the Cost of Equity for Stansfield Enterprises

Suppose the stock of Stansfield Enterprises, a publisher of PowerPoint presentations, has a beta of 1.5. The firm is 100% equity financed. Assume a risk-free rate of 3% and a market risk premium of 7%. What is the appropriate discount rate for an expansion of this firm?

Solution

  • The cost of equity (RS) is calculated as follows:
    • RS = RF + β × (RM − RF)= 3% + 1.5 × 7%= 13.5%
  • This 13.5% represents the return required by equity investors, serving as the discount rate for the firm’s potential projects.

Evaluating Investment Projects: An Example

Stansfield Enterprises is considering three independent projects, each with an initial cost of $100 and a duration of one year. The projects and their expected cash flows are as follows:

Note: All projects share the same beta (β) of 1.5, indicating a uniform level of systematic risk.

Using the cost of equity determined earlier (13.5%), let’s evaluate the financial viability of each project through their Internal Rate of Return (IRR) and Net Present Value (NPV):

Analysis:

  • Project A offers the highest return, significantly above the cost of capital, making it the most attractive investment.
  • Project B breaks even with the cost of capital, offering no additional value.
  • Project C yields a return below the cost of capital, making it an unattractive investment.

An all-equity firm should accept projects whose IRRs exceed the cost of equity capital and reject projects whose IRRs fall short of the cost of capital.

Determining the Risk-Free Rate

Identifying a risk-free rate is crucial for financial valuations:

  • In the U.S., Treasury bills and bonds are considered nearly risk-free due to the government’s backing.
    • For international valuations, opt for a local government (sovereign) bond in the currency of the valuation.
  • Match the maturity of the risk-free security with that of the cash flows for the firm or project:
    • Use a 20- or 30-year U.S. Treasury bond for long-term valuations.
  • Focus on the current yield of the bond:
    • This reflects the most accurate, up-to-date risk-free rate, avoiding reliance on historical averages.

Estimating the Market Risk Premium: Historical Data Approach

The historical data approach involves analyzing past returns to forecast future premiums:

  • Choice of Index: Use a market-value-weighted index or a portfolio of large-company stocks as a proxy for the market.
  • Time Series Length: Select between long-term historical averages for stability or shorter periods to capture recent trends.
  • Risk-Free Rate Comparison: Calculate the average return on risk-free securities over the same period to determine the premium.

This method relies on the assumption that historical market behaviors are indicative of future trends.

Estimating the Market Risk Premium: Dividend Discount Model

The Dividend Discount Model (DDM) offers a theoretical approach based on expected dividends⭐⭐⭐:

  • $R_M = D_1 / P_0 + g$

Implementing 【使用、实施】DDM: Utilize market data and analyst forecasts for next year’s dividends (D1), current stock prices (P0), and expected growth rates (g).

Forward-Looking Estimates: DDM calculates the expected market return by considering future dividend payments and growth, providing an insight into anticipated market performance.

  • This approach contrasts with historical analysis by focusing on future dividend prospects and growth expectations to estimate the market’s return.

Understanding and Estimating Beta

Beta ($β_i$) measures a stock’s return sensitivity to the market:

  • $βi = Cov(R_i, R_M) / Var(R_M) = σ_{i,M} / σ^2_M$

Key Concepts:

  • Market Portfolio: Ideally includes all assets in the economy. Practically, broad stock market indices like the S&P Composite serve as proxies for the market.
  • Beta (β): Indicates how much a stock’s return is expected to change relative to a change in the market return. A β greater than 1 suggests higher sensitivity to market movements, while a β less than 1 indicates lower sensitivity.

Understanding β is crucial for both portfolio management and capital budgeting, as it helps in assessing the risk and expected return of investments in relation to market fluctuations.

Understanding the Stability of Beta

Beta’s stability is key in assessing risk, but it’s subject to variations:

  • Industry Consistency: Betas tend to remain stable over time for firms within the same industry, reflecting consistent risk levels associated with industry-specific factors.
  • Factors Influencing Beta Variability: Despite its general stability, several factors can lead to changes in a firm’s beta:
    • Product Line Changes: Diversification or specialization in product offerings can alter risk exposure.
    • Technological Advances: Innovations may disrupt industry norms, affecting firm-specific risks.
    • Deregulation: Changes in regulatory environments can shift competitive landscapes, impacting risk.
    • Financial Leverage: Adjustments in a firm’s capital structure, particularly debt levels, can modify its risk profile.

Understanding these dynamics is crucial for investors and managers in making informed decisions regarding risk management and investment strategy.

Cyclicality of Revenues [收入的周期性]and Beta

Understanding beta involves recognizing the cyclical nature of a company’s revenues:

  • Revenue Cyclicality: Stocks in highly cyclical industries, like retail and automotive, tend to have higher betas, reflecting greater sensitivity to economic cycles.
  • For example, empirical evidence shows that the performance of retailers and automotive firms is closely tied to the business cycle.
  • Conversely, sectors such as transportation and utilities exhibit less sensitivity to economic fluctuations, often resulting in lower betas.
  • Cyclicality vs. Variability:
    • It’s crucial to distinguish between cyclicality and mere revenue variability. A company like a movie studio may experience significant revenue variability (hits vs. flops), yet its beta might not be high if its revenues are not deeply correlated with the broader economic cycle.
  • Beta, therefore, is a measure not just of volatility but of how a stock’s returns move in conjunction with market returns.

The Impact of Operating Leverage on Beta

Operating leverage illustrates the financial dynamics of fixed versus variable costs and their effect on a firm’s profitability and risk:

  • High Operating Leverage means a firm has significant fixed costs relative to variable costs, which amplifies earnings volatility in response to sales fluctuations.
  • Beta Sensitivity: This earnings volatility, driven by operating leverage, directly influences the firm’s beta. Higher operating leverage leads to a higher beta, indicating greater sensitivity to market movements.
  • DOL Formula: $DOL = \frac{∆EBIT}{EBIT}\times \frac{∆Sales}{Sales}$
    • This formula demonstrates how changes in sales (∆Sales) can affect the earnings before interest and taxes (EBIT). A higher DOL indicates that a smaller change in sales can lead to a larger change in EBIT, which in turn affects the beta.

Visualizing the Effect of Operating Leverage

The concept of operating leverage is further elucidated through visualization:

  • The relationship between operating leverage and beta can be further understood through visualization, where:
    • A higher DOL indicates a steeper slope in the EBIT-to-sales line, reflecting increased earnings volatility for a given change in sales.
    • This volatility translates to a higher beta, signifying a greater sensitivity of the firm’s equity to market movements.

The Impact of Financial Leverage on Beta

Financial leverage adds another layer of sensitivity to a firm’s financing costs:

  • Operating vs. Financial Leverage: While operating leverage relates to fixed production costs, financial leverage concerns fixed financing costs.

  • Beta Relationship: The beta of a firm’s assets (βAsset) is a weighted average of its equity beta ($β_{Equity}$) and debt beta ($β_{Debt}$), reflecting the overall risk of the firm:

    $$β_{Asset} = (S / (S + B)) × β_{Equity} + ((B / (S + B)) ×β_{Debt})$$

  • where S is the market value of equity, and B is the market value of debt.

  • Financial leverage increases the equity beta compared to the asset beta, amplifying equity risk due to the firm’s debt.

Simplifying Assumptions on Debt Beta

Considering the practical aspects of financial leverage:

  • Low Debt Beta: In reality, the beta associated with a firm’s debt (βDebt) is often minimal due to:
    • Debt’s seniority in the capital structure.
    • The relative stability of returns to debt holders.
  • Assuming βDebt ≈ 0 simplifies the relationship to:
    • βAsset = S / (S + B) × βEquity
  • This assumption leads to βAsset being lower than βEquity, illustrating how leverage inflates the perceived risk of equity relative to the entire firm.

Example: Financial Leverage and Beta

Asset vs. Equity Betas: Rapid Cedars, Inc.

  • Rapid Cedars, Inc., an all-equity tree-growing company, has an equity beta of 0.8. The company plans to adjust its capital structure to a ratio of one part debt to two parts equity. What implications does this have for the company’s asset beta and equity beta?

Analysis and Solution:

  • Asset Beta: Remains unchanged at 0.8, as the company’s operational risk does not alter with changes in financing.
  • Equity Beta Calculation: Assuming the debt has a beta of zero (common in practice due to its seniority and return stability), the equity beta adjusts to reflect the new financial risk:

Beta and Corporate Taxes: The Fundamental Relationship

  • The introduction of corporate taxes (τC) adjusts the beta dynamics due to the tax shield effect of debt:
    • $β_{Asset} = (S / (1 − τ_C)B) × β_{Equity} + (1 − τ_C)B / (1 − τ_C)B × β_{Debt}$
    • This equation highlights how debt financing, adjusted for corporate taxes, influences the firm’s overall risk profile.
    • The tax shield (1 − τC) effectively reduces the cost of debt, impacting the weighted risk (beta) calculations for assets and equity.
    • Understanding this relationship is crucial for accurately assessing the cost of capital in a taxed environment.

Simplifying Equity Beta with Tax Considerations

  • Assuming the beta of debt (βDebt) is zero, reflecting its relative stability and lower risk:
    • βEquity = βAsset × (1 + (1 − τC)B / S)
    • This simplification allows for a clearer understanding of how corporate taxes amplify the equity beta compared to the asset beta.
    • It shows that equity risk (beta) increases with financial leverage, especially after accounting for the tax benefits of debt.
    • The formula underscores the significance of leveraging and taxes in determining a firm’s equity risk relative to its operational risk.

Capital Budgeting & Project Risk

  • Using a single discount rate across all projects may inadvertently increase the firm’s risk and decrease its value:
    • Different risk profiles of projects necessitate distinct discount rates, tailored to their specific beta.
    • A uniform discount rate can lead to misallocation of capital—favoring riskier projects and neglecting safer, potentially valuable opportunities.
    • The Security Market Line (SML) offers insight into correctly pricing projects according to risk:
      • Projects above the SML are undervalued and may be wrongly rejected, despite promising positive NPV.
      • Conversely, projects below the SML are overvalued, potentially leading to acceptance of negative NPV investments.
      • The SML helps in determining the appropriate risk-adjusted discount rate, ensuring value creation aligns with the risk undertaken.

Choosing the Right Cost of Capital for Project Evaluation

  • Conglomerate Company has a diverse range of investment projects with varying levels of risk (beta). The company’s overall cost of capital is 11.1%, calculated using CAPM:
    • 11.1% = 2% + 1.3 × 7%
  • The breakdown of the company’s investment projects by sector is as follows:
    • Automotive Retailer: β = 2.0
    • Computer Hard Drive Manufacturer: β = 1.3
    • Electric Utility: β = 0.6
    • Given the average β of assets is 1.3,
    • Question: For a new electrical generation investment, reflecting the characteristics of the Electric Utility sector, which cost of capital should Conglomerate Company use?
    • Answer: R = 2% + 0.6 × 7% = 6.2%
    • 6.2% reflects the opportunity cost of capital on an investment in electrical generation, given the unique risk of the project.

Understanding the Cost of Debt

  • The cost of debt is a critical component of a firm’s capital structure:
    • It is typically measured by the interest rate required on new debt issuance, which is the yield to maturity on the company’s outstanding debt.
    • This rate must be adjusted for the tax deductibility of interest expense, which reduces the effective cost of debt for the firm.
    • The formula for the after-tax cost of debt is:
      • After-tax Cost of Debt = Interest Rate × (1 − Tax Rate)

Example Calculation of the Cost of Debt

  • Consider a comparison between an unlevered and a levered corporation to understand the impact of debt on after-tax earnings:
    • Unlevered Corp. | Levered Corp.
    • Revenue | $180.0 | Revenue | $180
    • Expenses | 70 | Expenses | 70
    • Pretax earnings | $110.0 | Earnings before interest and taxes | $110
    • Taxes (21%) | 23.1 | Interest (10% on $100 borrowed) | 10
    • Aftertax earnings | $86.9 | Pretax earnings | $100
    • Aftertax earnings | | Taxes (21%) | 21
    • Aftertax Cost of Debt | | Aftertax earnings | $79
    • After-tax Cost of Debt = 10% × (1 − 0.21) = 7.9%

Calculating the Cost of Preferred Stock

  • Preferred stock occupies a unique place in a company’s capital structure:
    • Characteristics: Preferred stock combines features of both bonds and common stock, typically paying a fixed dividend similar to a bond’s coupon payment.
    • Hierarchy: It ranks junior to debt but senior to common stock regarding claim on assets and earnings.
    • Cost Calculation: The cost of preferred stock (RP) can be determined by the dividend paid © divided by its current market price (PV ):
      • RP = C / PV

Introducing the Weighted Average Cost of Capital (WACC)

  • The WACC represents the average rate of return a company is expected to pay its security holders to finance its assets:
    • The general formula for WACC is:
      • WACC = (S / V) × RS + (B / V) × RB × (1 − τC)
    • Including preferred stock, the WACC formula adjusts to:
      • WACC = (S / V) × RS + (B / V) × RB × (1 − τC) + (P / V) × RP
    • Where P is the market value of preferred stock and RP is the cost of preferred stock.

WACC Example Calculation - Step by Step

  • Calculating WACC: A Step-by-Step Example

    • Consider a firm whose debt has a market value of $40 million and whose stock has

    • market value of $60 million. The firm pays a 5 percent rate of interest on its new debt and has a beta of 1.41. The corporate tax rate is 21 percent. Assume that the CAPM holds, the risk premium on the market is 9.5 percent, and the current Treasury bill rate is 1 percent. What is this firm’s WACC?

    • We will calculate:

      • The aftertax cost of debt (RB × (1 − τC))
      • The cost of equity using CAPM (RS)
      • The proportions of debt and equity based on market values

WACC Example Calculation - Results

  • 1 Aftertax cost of debt: 3.95% (5% ×(1 − 21%))
  • 2 Cost of equity using CAPM: 14.40% (1% + 1.41 × 9.5%)
  • 3 Proportions of financing: Debt 40%, Equity 60%
  • Components | MV ($ Million) | Weights | Cost | WCC
  • Debt | 40 | 0.4 | 3.95% | 1.58%
  • Equity | 60 | 0.6 | 14.40% | 8.64%
  • Total | 100 | 1.0 | 10.22%
  • The calculated WACC for the firm is 10.22%.

Valuing a Finite-Horizon Project

  • Suppose a firm has both a current and a target debt-equity ratio for a project of .6, a pretax cost of debt of 5.15 percent, and a cost of equity of 10%. The corporate tax rate is 21%. Now the firm is considering taking on a warehouse renovation costing $60 million that is expected to yield additional cash flows of $12 million a year for six years. The warehouse has a finite horizon and is expected to be worthless after Year 6. Assume the warehouse renovation project has the same risk as the firm. Should the firm accept the project?
  • Solution:
    • Our first step is to calculate the WACC of the firm.
    • A B/S ratio of .6 means the debt-value ratio is 6/(6 + 10) = .375 and the equity-value ratio is 10/(6 + 10) = .625.
    • The WACC will then be: WACC = 0.625 × 0.1 + 0.375 × 0.0515 × 0.79 = 0.0778.
    • Using the NPV equation and discounting the six years of expected cash flows from the renovation at the WACC, we have:
      • NPV = −$60 + $12 / (1 + WACC)^1 + · · · + $12 / (1 + WACC)^6
      • = −$60 + $12 × (1 − (1 + 0.0778)^6) / 0.0778
      • = −$60 + $12 × 4.0601
      • = −$4.15
    • The firm should reject the project.

Estimating WACC in Practice*

  • We will now calculate the cost of capital for a real company, Eastman Chemical Co.
  • A leading international chemical company and maker of plastics for soft drink containers and other uses.
  • Created in 1993 from Eastman Kodak.
  • First, we estimate the cost of equity and the cost of debt.
  • We estimate an equity beta to estimate the cost of equity.
  • We can often estimate the cost of debt by observing the YTM of the firm’s debt.
  • Second, we determine the WACC by weighting these two costs appropriately.

Eastman’s Beta*

  • Regress five years of monthly stock returns on the market return.
  • Look up online. (Reuters, Yahoo Finance)
  • Source: Yahoo Finance, as of 2023/03/26.

Eastman’s Cost of Equity*

  • We assume a market risk premium of 5.94%.
  • From Aswath Damodaran’s estimation.
  • Our estimate of the risk-free rate is the current Treasury rate of 3.38%.
  • Using Eastman’s beta in the CAPM to estimate the cost of equity,
    • RS = 0.0338 + 1.50 × 0.0594 = 0.1229.
  • The market capitalization is $9.41 billion as of 2023/03/26.

Eastman’s Cost of Debt*

Eastman’s Cost of Debt* (Continued)

  • Coupon Rate | Maturity | Market Value | Percentage of Total | YTM | Weighted YTM
  • 5.750 | 03/08/2033 | 500 | 15.31% | 5.268 | 0.81%
  • 7.600 | 02/01/2027 | 196 | 6.52% | 4.031 | 0.26%
  • 4.800 | 09/01/2042 | 500 | 13.09% | 5.784 | 0.76%
  • 3.800 | 03/15/2025 | 700 | 20.31% | 4.778 | 0.97%
  • 7.250 | 01/15/2024 | 198 | 5.89% | 6.048 | 0.36%
  • 6.750 | 06/15/2024 | 43 | 1.32% | 4.575 | 0.06%
  • 4.650 | 10/15/2044 | 900 | 22.98% | 5.759 | 1.32%
  • 4.500 | 12/01/2028 | 500 | 14.56% | 4.792 | 0.70%
  • Total | 3383.64 | 100.00% | | 5.24%

Eastman’s WACC*

  • The total value of the firm is $12.794 billion (= $3.384 + $9.41).
  • The debt percentages is 26.45% (= 3.384/12.794).
  • The equity percentage is 73.55% (= 9.41/12.794).
  • Assuming a tax rate of 21 percent, Eastman’s WACC is:
    • WACC = 0.2645 × 0.0524 × 0.79 + 0.7355 × 0.1229 = 0.1013.

Understanding Flotation Costs

  • Flotation costs are critical to understanding the true cost of raising new capital:
    • These costs cover the expenses associated with issuing new bonds or stocks.
    • They effectively increase the initial outlay of a project, thus impacting the Net Present Value (NPV) by:
      • Amount Raised = Necessary Proceeds / (1 − Flotation Cost Percentage)
    • The weighted average flotation cost (fA) depends on the proportion and cost of each component of the capital structure:
      • fA = (E / V) × fE + (D / V) × fD
    • where E is equity, D is debt, V is the total financing, fE is the flotation cost for equity, and fD is the flotation cost for debt.

Flotation Costs Example Calculation

  • Calculating True Cost with Flotation Costs
    • The Weinstein Corporation needs to raise $65 million with an 80% equity and 20% debt capital structure. Flotation costs are 20% for equity and 6% for debt. What is the total amount required, including flotation costs?
    • Solution:
      • First, calculate the weighted average flotation cost (fA):
        • fA = 0.80 × 0.20 + 0.20 × 0.06 = 0.172
      • Next, determine the total amount to be raised to cover the flotation costs:
        • Total Cost = $65m / (1 − 0.172) ≈ $78.5m
      • Including flotation costs, Weinstein must raise approximately $78.5 million.

Key Takeaways - Cost of Capital

  • We have explored several critical aspects of capital costs:

    • Dividends vs. Reinvestment: Firms must carefully choose between distributing excess cash as dividends or reinvesting it in capital projects.

    • Beta’s Role: The expected return on an investment is closely tied to its beta, which reflects the investment’s risk relative to the market.

    • Asset and Equity Beta: While asset beta measures market sensitivity of a firm’s assets, equity beta additionally accounts for financial leverage.

    • Project-Specific Discount Rates

    • Project-Specific Discount Rates: Projects with betas different from the firm’s overall beta should be evaluated using a discount rate reflective of the project’s unique risk.

    • WACC for Mixed Financing: When a project is financed with both debt and equity, the Weighted Average Cost of Capital (WACC) is the appropriate discount rate.

    • Understanding the cost of capital is crucial for making informed investment decisions and for the overall financial health of a company.

The End

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